# Lattice pricing of derivatives under multi curve framework (OIS and LIBOR)

My goal is to price various derivatives resetting to 1M and 3M LIBOR via using a lattice. I have calculated an OIS curve for discounting and adjusted 1M and 3M LIBOR forward curves to be consistent with OIS discounting and current LIBOR/basis swap rates.

Is there a way to go about doing this (with possibly some simplifying assumptions) with a short rate model (Hull-White in this case) which has:

1. Analytical formulas for caplet and swaption prices for the calibration step.
2. Only requires a 1 parameter short rate lattice to be built for derivative pricing (of say Bermudan swaptions of some sort).

I'd be happy to get some reference(s) for formulas in (1), and perhaps some advice on (2). I'm wondering if there is a way to use market vols to produce a short rate lattice implying the correct forward rates for setting up the cash flow, then build another short rate lattice for discounting via adjusting the first lattice uniformly downward at each tenor to match the OIS discount curve. Any ideas on how one could attack this would be helpful!

There are many resources describing how to build a trinomial tree for the Hull & White model (for instance http://www-2.rotman.utoronto.ca/~hull/downloadablepublications/TreeBuilding.pdf), and finite differences schemes are popular as well. These apply to the single curve case.

To deal with the multi curve case while keeping everything 1 factor, the usual way is

• to assume that the Hull & White model short term rate $r$ represents the OIS short rate, and to build the OIS single curve model accordingly ;
• and to assume that the OIS-Libor basis is deterministic and evolves along its forward value.

For instance if $D_{\text{OIS}}(0, T)$ is the initial OIS zero curve and $D_{\text{3M}}(0, T)$ is the initial 3M Libor zero curve, then the initial basis zero curve is $D_{\text{OIS-3M}}(0, T) = D_{\text{3M}}(0, T) / D_{\text{OIS}}(0, T)$. The assumption that the basis evolves deterministically means that at any time $t$ in the future you have $D_{\text{OIS-3M}}(t, T) =D_{\text{OIS-3M}}(0, T)/D_{\text{OIS-3M}}(0, t)$, or written differently $$D_{\text{3M}}(t, T)=D_{\text{OIS}}(t, T)\times (D_{\text{OIS-3M}}(0, T)/D_{\text{OIS-3M}}(0, t))$$

From this assumptions all the closed form formulas in the single curve Hull & White model (for caps/floors, European swaptions, bond options) translate into modified closed form formulas in the multi curve framework so calibration is still efficient. Likewise Bermuda swaptions are priced off the trinomial or other discretization scheme.

• Thank you Antoine! I am wondering if your suggestion is equivalent to the assumption that the OIS short rate and 3M LIB short rate differ by a deterministic function? $$r^{OIS} (t) + \phi(t) = r^{3M}(t)$$ Then in terms of bond prices we should get: \begin{align*} P^{OIS}(t,T) &= \widetilde{\mathbb{E}} [ e^{-\int_t^T r^{OIS}(u)du} | \mathcal{F}_t ] \\ &= e^{\int_t^T \phi(u) du} \widetilde{\mathbb{E}} [e^{-\int_t^T r^{3M}(u)du} | \mathcal{F}_t ] \\ &= \xi (t,T) P^{3M} (t,T) \end{align*} Here $\xi$ would be $D_{OIS-3M}$? – RyanM May 4 '18 at 14:25
• Yes that is exactly right. – Antoine Conze May 4 '18 at 14:28